GLOBAL navigation satellite system reflectometry

Transcription

1 1064 IEEE GEOCIENCE AND REMOTE ENING LETTER, VOL. 13, NO. 8, AUGUT 2016 On the patial Resolution of GN Reflectometry Maria Paola Clarizia, Member, IEEE, and Christopher. Ruf, Fellow, IEEE Abstract A method for defining the spatial resolution of a Global Navigation atellite ystem reflectometry delay Doppler map DDM) and of any derived geophysical product is proposed. An effective spatial resolution is derived as a function of measurement geometry and delay Doppler DD) interval, and as a more appropriate representation of resolution than the geometric resolution previously used in the literature. The definition more accurately accounts for variations in the scattered power across different pixels of the DDM and more accurately includes the power spreading effect caused by the Woodward ambiguity function. The dependence of the effective resolution on incidence angle, receiver altitude, and DD interval is analyzed and compared with the dependence of the geometric resolution with similar parameters. Index Terms Delay Doppler map DDM), Global Navigation atellite ystem reflectometry GN-R), resolution. I. INTRODUCTION GLOBAL navigation satellite system reflectometry GN-R) is recognized as a promising remote sensing technique that fulfills the need for high spatial coverage, short temporal revisit time, and reduction in cost for satellite missions. The most successful application of GN-R for ocean remote sensing so far is the retrieval of surface roughness and near-surface wind speed [1] [5]. The typical algorithms used to estimate the wind and wave parameters in scatterometric GN-R can be divided into two categories: 1) approaches that exploit the delay Doppler map DDM) over a selected delay Doppler DD) window, centered around the specular point, to retrieve the geophysical parameters of interest, usually through the definition of an observable [3] [5] or by fitting a model to the DDM [2]; and 2) approaches that perform a DDM deconvolution and a DD-to-space inversion to obtain an image of the normalized radar cross section [7] and, hence, of the wind speed or wave field over a given area. The first type of algorithm is more commonly used due to the difficulty in deconvolving the DDM and in resolving the space-to-dd ambiguity [1]. However, assigning a spatial resolution to the geophysical parameters estimated from a particular DD region presents challenges. One previous wind speed retrieval study assigned a spatial resolution to their wind product that is purely based on geometrical considerations [5]. Consideration on how to determine the spatial resolution is discussed in [6] without a specific mathematical definition. Manuscript received November 30, 2015; revised March 3, 2016; accepted April 29, Date of publication June 2, 2016; date of current version July 20, The authors are with the Department of Climate and pace ciences and Engineering, University of Michigan, Ann Arbor, MI UA. Color versions of one or more of the figures in this paper are available online at Digital Object Identifier /LGR Fig. 1. a) Geometric case 1: When the iso-doppler lines at the minimum and maximum Doppler frequencies of the selected interval red parabola) do not cross the iso-delay line black ellipse) at the maximum delay of the selected interval. b) Geometric case 2: When the iso-doppler parabola crosses the isodelay ellipse. In both cases, the geometric resolution is the square root of the gray area. Here, a definition of effective spatial resolution is proposed, which expands on the geometric resolution i.e., the area at the intersection of the maximum iso-delay and iso-doppler lines of the DD window [5], [8]) by incorporating the effect of varying scattered power across the physical area and the spreading effect of the Woodward ambiguity function WAF). For simplicity, we focus on GP signals with the WAF illustrated in [9]. Our definition can be extended to other types of navigation signals with different WAFs. The effective spatial resolution is described as a function of geometry i.e., receiver altitude and incidence angle) and a size of the DD interval, and can be used to assign a spatial resolution to any geophysical parameter retrieved from a DDM. This letter is organized as follows. ection II summarizes the definition of geometric resolution and presents a formal mathematical definition of the effective spatial resolution. ection III illustrates the dependence of effective spatial resolution on incidence angle see ection III-A), receiver altitude see ection III-B), and DD interval see ection III-C), and compares the effective spatial resolution with geometric spatial resolution in all three cases. ection IV presents a compact polynomial approximation of the full formulation of the effective spatial resolution, and ection V summarizes the conclusions of this letter. II. DEFINITION OF PATIAL REOLUTION The geometric spatial resolution corresponding to a given DD interval in GP-reflectometry GP-R) has been defined in [5] as the square root of the spatial scattering area within the DD interval considered, i.e., the area at the intersection between the iso-delay ellipse at the maximum delay considered, and the two iso-doppler parabolas at the minimum and maximum Doppler frequencies considered. Two possible configurations of iso-delay ellipses and iso-doppler lines are illustrated in Fig. 1, for a given geometry and DD interval X 2016 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. ee for more information.

2 CLARIZIA AND RUF: ON THE PATIAL REOLUTION OF GN-R 1065 Fig. 2. a) Weighting function and b) physical scattering area for h 0 = 500 km, θ =20, =1chip, and fm D = 2000 Hz. Hence, the geometric spatial resolution can be either determined by the iso-delay ellipse only when the iso-doppler lines fall outside the area delimited by the ellipse see case 1 in Fig. 2) or by the actual intersection of iso-delay ellipse and iso- Doppler lines see case 2 in Fig. 2). This definition of spatial resolution relies purely on geometrical considerations and does not take into account: 1) the variations in the received power across the different points within the DD window boundaries; and 2) the residual scattered power from points immediately outside the DD window boundaries, as a result of the spreading caused by the WAF. An alternate effective spatial resolution is proposed, whose definition takes into account both the aforementioned effects. The total received power P tot over a given DD interval can be expressed as P tot = P DDM h0,θ,τ l,fm where P DDM is the received power in a single pixel of the DDM; h 0 is the receiver altitude; θ is the incidence angle see Fig. 7 in the Appendix); τ l and f D m are the delay and Doppler coordinates, respectively; and the summation extends over the L discrete delay pixels and 2M +1discrete Doppler pixels that form the selected DD interval. The individual DDM pixels can be expressed as P DDM h0,θ,τ l,fm = σ 0 x, y, h 0,θ)W x, y, h 0,θ,τ l,fm) D 2) where the integral is over the portion of the Earth surface forming the glistening zone, and W...) is a weighting function comprised of all the other terms in the bistatic GP scattering equation [9]. The weighting function can be written as W x, y, h 0,θ,τ l,fm G R = K RT 2 WAF 2 [ t, τ l,f D,f D ] R2 m R 3) where a) K = λ 2 P T G T /4π) 3 with λ as the GP wavelength and G T and P T being, respectively, the transmit antenna gain and power; b) t = tx, y, θ, h 0 ) and f D = f D x, y, θ, h 0 ) are the delay and Doppler frequency coordinates corresponding to the space coordinates x, y); 1) c) G R = G R x, y, θ, h 0 ) is the receive antenna gain; d) R T = R T x, y, θ, h 0 ) and R R = R R x, y, θ, h 0 ) are the distances from the scattering point on the glistening zone at x, y, z) to, respectively, the transmitter and receiver; e) and WAF is the Woodward ambiguity function, expressed in [9]. Our definition for effective spatial resolution is adapted from a definition of effective bandwidth commonly used in the filter theory [10]. We consider the summation of the weighting function over the entire DD window, i.e., I W = W x, y, h 0,θ,τ l,fm) D 4) and we consider an equivalent quantity given by the following product: I W = R2 eff h0,θ,,fm Wavg h0,θ,,fm) D 5) where the average weighting function is given by W avg h0,θ,,fm W ) ) x,y,h 0,θ,τ l,fm D Π τl /2 ) Π f D m D = ) Π τl /2 ) Π f D m D 6) and the functions Π are rectangular functions. We impose I W = I W and define the effective spatial resolution as R eff h0,θ,,fm W x, y, h 0,θ,τ l,fm) D = ) W avg h0,θ,,fm D. 7) For brevity, we omit the dependence of R eff on h 0,θ,,fM D). By combining 6) and 7), we rewrite R eff as 8), shown at the bottom of the next page, where R GEO is the geometric resolution, given by R geo h0,θ,,fm = ) ) τl /2 f D Π Π m D. 9) The expression for R eff can be made more explicit by writing each term in W...) as a function of the incidence angle θ, the receiver altitude h 0, and the DD region considered, delimited by 0and in delays, and by fm D to f M D in Doppler frequency, as illustrated in the Appendix. The determination of R eff...) is performed by defining the geometry and DD parameters θ, h 0,and, and by calculating

3 1066 IEEE GEOCIENCE AND REMOTE ENING LETTER, VOL. 13, NO. 8, AUGUT 2016 Fig. 3. a) Effective resolution and b) ratio between effective and geometric resolutions as a function of the incidence angle, for h 0 = 500 km, and two DD intervals of, respectively, black stars) chips/ Hz and white circles) 0 1 chips/ Hz. the effective spatial resolution using 3), 8), and 9). Note that the effective spatial resolution should also ideally depend on the azimuth angle i.e., the angle formed by the receiver velocity vector and the x-axis of the specular frame) through the orientation of the receiver velocity vector. However, this dependence is, in practice, small compared with the dependence on incidence angle and thus neglected here. Fig. 2 shows an example of the weighting function and physical area for a specified receiver altitude, incidence angle, and DD interval. A spherical Earth approximation has been used to determine the z-coordinate z i of the surface points, and the receiver antenna gain G R...) has been calculated using the Cyclone Global Navigation atellite ystem CYGN) 2-D antenna pattern see Appendix). Fig. 2 highlights how different weights are assigned to the locations contributing to the scattered power and how the actual area covered by the weighting function is larger than the geometric area. However, the two areas become closer for increasing the DD interval, as described in ection III. III. DEPENDENCE OF R eff ON GEOMETRY AND DD PARAMETER Variations in the effective spatial resolution as a function of the incidence angle, the receiver altitude, and the DD interval are considered, and the dependencies are compared with those of the geometric resolution. A. Dependence on the Incidence Angle The effective spatial resolution is shown as a function of the incidence angle in Fig. 3a) for two different DD intervals. Fig. 3b) illustrates the dependence of the ratio between effective and geometric resolutions on incidence angle, for the same DD intervals as in Fig. 3a). Fig. 3a) demonstrates that the effective spatial resolution increases with increasing incidence angle, similarly as the geometric resolution [5]. The similarity of their dependence on incidence angle is confirmed Fig. 4. a) Effective resolution and b) ratio between effective and geometric resolutions as a function of the receiver altitude, for θ =20 and the same DD intervals in Fig. 3. by Fig. 3b), where the ratio exhibits little variation with respect to incidence angle for both the DD intervals considered. B. Dependence on Attitude The effective spatial resolution is also considered for varying receiver altitudes, assuming a constant incidence angle and the same DD intervals as the aforementioned. A plot of R eff as a function of altitude is shown in Fig. 4a). The effective resolution increases with altitude, as a result of the stretching of the iso-delay and iso-doppler lines, i.e., the same DD interval covers a larger area. The geometric resolution is characterized by the same type of increase, and no significant difference is noted between their dependence on altitude, as shown in Fig. 4b). C. Dependence on DD Intervals The effective resolution is considered at a fixed geometry 20 incidence angle, receiver altitude of 500 km) and for different DD intervals. The delay interval always starts at the zero chip delay i.e., delay relative to the specular point) and its upper bound varies from chips, with a step size of 0.25 chips. The Doppler interval varies from [ ] to [ ] Hz, with a step size of ±500 Hz for the lower and upper Doppler bounds. Fig. 5a) and b) shows the 2-D map of effective resolution and ratio between effective and geometric resolutions as a function of the delay and Doppler intervals. Fig. 5a) and b) shows that the geometric resolution is lower than the effective resolution at the small DD interval, whereas they tend to become closer for increasing DD intervals, which is confirmed by the tendency of their ratio to become close to unity in Fig. 5b). The stronger differences for the small delay and Doppler intervals occur because the additional area, introduced by the weighting function at the edges of the geometric area, is usually a larger proportion of the geometric area itself when this area is small and reduces as the geometric R eff = W x, y, h 0,θ,τ l,fm) D ) W x, y, h 0,θ,τ l,fm D)Π τl /2 Π f D m D ) R geo 8)

4 CLARIZIA AND RUF: ON THE PATIAL REOLUTION OF GN-R 1067 TABLE I COEFFICIENT FOR THE POLYNOMIAL FIT OF THE FOURTH DEGREE OF fh 0,θ) AND g,fm.the VARIABLE x AND y IN THE POLYNOMIAL EQUATION AT THE TOP OF THE TABLE REPREENT fm D AND τ l IN THE CAE OF g), AND h 0 AND θ IN THE CAE OF f) Fig. 5. a) Effective resolution in kilometers and b) ratio between effective and geometric resolutions as a function of DD interval. Fig. 6. Ratio of effective to geometric resolution versus geometric resolution for h 0 = 500 km, θ =20, and all the different DD intervals shown in Fig. 5. Cases 1 and 2 refer to the configuration of iso-delay and iso-doppler lines illustrated in Fig. 1. area gets larger. This is, however, strictly true only when the iso-doppler lines at the minimum and maximum Doppler of the interval considered are outside the iso-delay ellipse at the maximum delay, i.e., when case 1 of Fig. 1 occurs. Variations in the geometric and effective areas in the two cases highlighted in Fig. 1 are illustrated in Fig. 6, where the ratio of effective resolution to geometric resolution is plotted versus the geometric resolution and color coded for cases 1 and 2. Fig. 6 shows a monotonic decrease in the ratio of effective resolution to geometric resolution with increasing geometric resolution for case 1, with the ratio converging to unity for large areas. This happens when the resolution is determined only by the size of the iso-delay ellipse at the maximum delay, for which an elliptical annulus is introduced by the weighting function as an additional scattering area, and the width of this annulus decreases with increasing delay interval, due to the iso-delay ellipses becoming closer to each other [1]. The relationship is more complicated in case 2, when the iso-doppler lines cut across the iso-delay ellipse. In this case, the additional area introduced by the weighting function is a subset of the annulus, plus some additional area at the edges of the iso-doppler segments cutting through the iso-delay ellipse. Variations in this area with increasing DD interval is not monotonic and depends on the particular DD interval considered, as shown in the behavior of the red points in Fig. 6. IV. POLYNOMIAL FORMULATION OF EFFECTIVE PATIAL REOLUTION The effective spatial resolution is a fairly complex function of four parameters, and as such, its numerical implementation can be lengthy and cumbersome. However, the dependence of the effective spatial resolution on the DD parameters and fm D is found to be relatively independent of the geometric parameters θ and h 0. This separability allows for an approximation of the effective spatial resolution to be formulated as the product of two functions as follows: R eff h0,θ,,fm = Rc eff fh 0,θ) g,fm 1 R c = g =0.25chip,fM D = 500 Hz). 10) The two functions fh 0,θ) and g,fm can be well approximated by polynomial functions of the fourth degree. The resulting equation and coefficients obtained by a least-squares fit are given in Table I. They have been obtained assuming the CYGN receiver antenna pattern, and they are strictly valid for delay ranges smaller than 10 chips. The fourth-degree best-fit polynomial functions produce an overall rms error of 0.08 km in case 1 and 0.57 km in case 2. Equation 10) is a useful tool for a quick estimation of the effective spatial resolution, but it represents an approximation and as such produces an error. The error between the true and approximate values of the effective spatial resolution has been calculated as a percentage of the true value, for a variety of geometries and receiver altitudes, and for the two DD pairs with the values of, respectively, [0 0.75] chip/[ ] Hz, and [0 1.5] chip/[ ] Hz. This error can be as low as 0.5% for the smaller DD interval and the lowest altitude case analyzed 200 km), and increases up to 12% for the larger DD interval and the higher altitudes because the true effective resolution is itself larger in these cases, i.e., P x, y) =a 1 + a 2 x + a 3 y + a 4 x 2 + a 5 xy + a 6 y 2 + a 7 x 3 + a 8 x 2 y + a 9 xy 2 + a 10 y 3 + a 11 x 4 + a 12 x 3 y + a 13 x 2 y 2 + a 14 xy 3 + a 15 y 4.

5 1068 IEEE GEOCIENCE AND REMOTE ENING LETTER, VOL. 13, NO. 8, AUGUT 2016 The delay and Doppler shifts t i and f i associated with a point on the surface P i =[x i,y i,z i ] can be written as Fig. 7. Geometry and specular frame coordinate system. R e is the Earth radius. V. C ONCLUION In this letter, we have presented a new definition of spatial resolution for DDM, called effective spatial resolution, following an approach similar to the definition of effective bandwidth that is typically used in the filter theory. An increase in the effective spatial resolution is observed for increasing incidence angle, receiver altitude, and DD window. The effective resolution is also compared with the geometric resolution, and the two exhibit the same general behavior with respect to changing measurement geometry and DD interval. The ratio between effective and geometric resolutions decreases with increasing geometric resolution, with the ratio tending toward unity at high > 60 km) geometric resolution and for large DD intervals, in particular when the iso-doppler lines at the minimum and maximum Doppler frequency of the selected interval fall outside the iso-delay ellipse at the maximum delay. A concise polynomial approximation for the effective spatial resolution is also derived. It can be used as a convenient means for determining the effective spatial resolution. The effective spatial resolution represents a more rigorous expression for the surface scattering area that contributes to the measurements in a DDM and in any geophysical product obtained from it. APPENDIX Here, explicit expressions for the terms in the weighting function W...) in 3) are given. We assume that the transmitter, receiver, and specular point coordinates are expressed in a specular point frame, as illustrated in Fig. 7. The transmitter has coordinates P t =[ H tanθ), 0,H + R e ], and receiver P r =[h tanθ), 0,h+R e ],whereh and h are related to the transmitter and receiver altitudes H 0 and h 0 by H= R e+ Re 2 +H0 2+2H 0R e +H0 2 tan2 θ)+2h 0 R e tan 2 θ) 1+tan 2 θ) h= R e+ Re 2 +h h 0R e +h 2 0 tan2 θ)+2h 0 R e tan 2 θ) 1+tan 2. θ) A1) t i = 1 c [R T x i,y i )+R R x i,y i )] [ f i = L 1 V t P t P i )/R T V r P r ] P i )/R R /c A2) where L 1 is the GP frequency; c is the speed of light, R T and R R are the ranges from transmitter/receiver to the point P i,and V t =[Vt x V y t Vt z] and V r =[Vr x V r y V r z ] are, respectively, the transmitter and receiver velocity vectors in the specular frame. The receiver antenna gain G R x i,y i ) is a function of elevation angle ω i and azimuth angle ϕ i of the point P i in the satellite orbit frame; hence, we need to relate ω i and ϕ i to x i and y i. The elevation and azimuth angles in the orbit frame for P i on the surface can be written as V r δi ω i =tan 1 Nc 2 α 2 i + β2 i ) ϕ i =tan 1 β i N c α i α i = Vx r x i h tanθ)) + Vy r y i + Vz r z i h R e ) β i = Vy r R e + h)x i h tanθ)) + + y i [ Vz r h tanθ) V x r R e + h)] + Vy r z i h R e )h tanθ) δ i = h tanθ)x i h tanθ)) + z i h R e ) h R e ) N c = h 2 tan 2 θ)+ h R e ) 2. A3) REFERENCE [1]. Gleason,. Hodgart,. Yiping, C. Gommenginger,. Mackin, M. Adjrad, and M. Unwin, Detection and processing of bistatically reflected GP signals from low Earth orbit for the purpose of ocean remote sensing, IEEE Trans. Geosci. Remote ens., vol. 43 no. 6, , Jun [2] O. Germain et al., The Eddy experiment: GN-R speculometry for directional sea-roughness retrieval from low altitude aircraft, Geophys. Res. Lett., vol. 31, no. 21, Nov. 2004, Art. no. L [3] J. Marchan-Hernandez et al., ea-state determination using GN-R data, IEEE Geosci. Remote ens. Lett., vol. 7, no. 4, , Oct [4] M. P. Clarizia, C.. Ruf, P. Jales, and C. Gommenginger, paceborne GN-R minimum variance wind speed estimator, IEEE Trans. Geosci. Remote ens., vol. 52, no. 11, pp , Nov. 2014, doi: /TGR [5] M. P. Clarizia and C.. Ruf, Wind speed retrieval algorithm for the Cyclone Global Navigation atellite ystem CYGN) mission, IEEE Trans. Geosci. Remote ens., vol. 54, no. 8, pp , Aug [6] V. U. Zavorotny,. Gleason, E. Cardellach, and A. Camps, Tutorial on Remote ensing Using GN bistatic radar of opportunity, IEEE Geosci. Remote ens. Mag., vol. 2, no. 4, pp. 8 45, Dec. 2014, doi: /MGR [7] E. Valencia et al., Ocean surface s scattering coefficient retrieval by Delay Doppler Map inversion, IEEE Geosci. Remote ens. Lett., vol. 8. no. 4, , Jul [8]. Gleason, C. Ruf, M. P. Clarizia, and A. O Brien, Calibration and unwrapping of the normalized scattering cross section for the cyclone global navigation satellite system, IEEE Trans. Geosci. Remote ens., vol. 54, no. 5, pp , May 2016, doi: /TGR [9] V. Zavorotny and A. Voronovich, cattering of GP signals from the ocean with wind remote sensing applications, IEEE Trans. Geosci. Remote ens., vol. 38, no. 2, , Mar [10] D. E. Johnson, Introduction to Filter Theory. Englewood Cliffs, NJ, UA: Prentice-Hall, 1976.